In rememberingsomer.comematics, dilation is a kind of transformation in i m sorry the size of a shape or geometric number is changed, but the loved one proportions and also shape remain the same. Below are numerous examples.

You are watching: If triangle abc is dilated by a scale factor of 3

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The quadrilateral and also cube above are dilated to produce smaller execution of the particular shapes. The model plane to the appropriate is dilated come one-thousandth the dimension of the genuine plane.

The ax "preimage" is offered to describe a geometric figure before it has actually been transformed and also the term "image" is offered after it has been transformed. Dilation is a non-rigid transformation, which means that the distance in between points in the preimage and also image don"t remain the same; the preimage and also image are not congruent. Dilation can refer come either rise or a decrease in size.

Scale factor

A scale aspect is a number by which a amount is multiplied, an altering the magnitude of the quantity. In the paper definition of dilation, the scale element is the value that identify both even if it is the preimage boosts or to reduce in size, as well as the magnitude of the change with respect come a fixed point called the facility of dilation. A facility of dilation have the right to either be part of the preimage or outside of the preimage. Listed below are some examples of both cases.

Scale factor in between 0 and also 1.

If the scale aspect is in between 0 and also 1, the preimage reduce in size.

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△ABC~△DEF

The preimage that triangle abc is dilated through respect to suggest O by a scale variable of to develop the picture of triangle DEF. The length of each side that the preimage is multiplied by to develop each corresponding side the the image. Also, point out on triangle DEF are the street from O loved one to their matching points top top triangle ABC.

Scale factor greater 보다 1

If the scale aspect is greater than 1, the preimage increases in size.

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△ABC~△GHI

Triangle alphabet is dilated v respect to suggest O by a scale aspect of to create triangle GHI. The size of each side of the preimage is multiplied by to produce each matching side the the image. Also, point out on triangle GHI are the street from O loved one to their corresponding points on triangle ABC.

Where go the image "land"?

When the scale element is in between 0 and also 1, the photo is in between the preimage and the facility of dilation. As soon as the scale variable is higher than 1, the preimage is between the image and also the center of dilation.

Below is a diagram mirroring a composite that the 2 dilations of triangle ABC.

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△ABC~△DEF~△GHI

Triangle DEF lies in between the preimage that triangle alphabet and point O, the facility of dilation, due to the fact that the scale element was less than 1. Triangle abc lies in between the dilated triangle GHI and also the facility of dilation because the scale factor was greater than 1. Also, all three triangles space similar.

The closer the scale factor is to 1, the closer the photo of the dilated triangle is to the preimage. Triangle abc is dilated by a scale factor of

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to develop triangle DEF:

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How to determine the scale factor

The scale variable can be identified by acquisition the ratio of the distance in between a allude on the image and the facility of dilation come the distance in between a corresponding point on the preimage and also the center of dilation.


Example:

Find the scale variable if quadrilateral ABCD is dilated to create quadrilateral EFGH because that the center of dilation O below.

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This method that the lengths the the sides of quadrilateral EFGH are

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the lengths the the corresponding sides of quadrilateral ABCD; the contrary ABCD is
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the size of EFGH.


Dilations in coordinate geometry

In a 2D coordinate plane, a dilation with origin as the dilation center and a scale aspect of "a" will maps a point (x, y) come (ax, ay). The following are part examples.

Center the dilation exterior of the geometric figure

Triangle alphabet is dilated by a aspect of 0.5 to develop triangle DEF. The coordinates of the vertices of triangle ABC room A (2, 6), B (4, 6), and C (4, 2). Dilating triangle alphabet by a aspect of 0.5 results in triangle DEF through vertices D (1, 3), E (2, 3), and F (2, 1).

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Center that dilation within of the geometric figure

Triangle abc is dilated by a aspect of 2 to produce triangle DEF. The coordinates of the vertices of triangle ABC are A (-2, -3), B (-3, 2), and also C (3, -1). Dilating triangle alphabet by a variable of 2 results in triangle DEF with vertices D (-4, -6), E (-6, 4) and also F (6, -2).

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Negative range factors

When the scale variable is negative, the beginning lies between the preimage and the image. The dimension of the photo is the same as that is once the scale aspect is positive, however is graphed in opposing direction.

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Triangle alphabet is dilated through a scale variable of -2 to produce triangle DEF. The works with of the vertices of triangle ABC space A (1, 3), B (2, 3) and C (2, 1). Dilating triangle abc by a scale aspect of -2 results in triangle DEF through vertices D (-2, -6), E (-4, -6), and F (-4, -2).

Center of dilation is no the origin

Let (c, d) be the coordinates of the facility of dilation. Offered that a is a scale element that is better than zero, the suggest (x, y) is dilated to a suggest (x", y"), as displayed below.

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To identify the collaborates of the dilated point, include the horizontal and vertical ranges to the x- and y-coordinates of the facility of dilation:

(x", y") = (c + a(x - c), d + a(y - d))


Example:

Dilate triangle ABC given vertices A (-1, 0), B (-1, 3), and also C (1, -3), scale element of 2, and a center of dilation of O (-4, -5).

Using the formula indigenous above, us can find the collaborates for the brand-new vertices D, E, and F and also graph them as follows:

D =(-4+2(-1-(-4)), -5+2(0-(-5)))
=(2, 5)
E =(-4+2(-1-(-4)), -5+2(-3-(-5)))
=(2, -1)
F =(-4+2(1-(-4)), -5+2(-3-(-5)))
=(6, -1)
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